Divide the following complex numbers. $ \dfrac{-7+22i}{2+3i}$
Answer: We can divide complex numbers by multiplying both numerator and denominator by the denominator's complex conjugate , which is ${2-3i}$ $ \dfrac{-7+22i}{2+3i} = \dfrac{-7+22i}{2+3i} \cdot \dfrac{{2-3i}}{{2-3i}} $ We can simplify the denominator using the fact $(a + b) \cdot (a - b) = a^2 - b^2$ $ \dfrac{(-7+22i) \cdot (2-3i)} {(2+3i) \cdot (2-3i)} = \dfrac{(-7+22i) \cdot (2-3i)} {2^2 - (3i)^2} $ Evaluate the squares in the denominator and subtract them. $ \dfrac{(-7+22i) \cdot (2-3i)} {(2)^2 - (3i)^2} = $ $ \dfrac{(-7+22i) \cdot (2-3i)} {4 + 9} = $ $ \dfrac{(-7+22i) \cdot (2-3i)} {13} $ Note that the denominator now doesn't contain any imaginary unit multiples, so it is a real number, simplifying the problem to complex number multiplication. Now, we can multiply out the two factors in the numerator. $ \dfrac{({-7+22i}) \cdot ({2-3i})} {13} = $ $ \dfrac{{-7} \cdot {2} + {22} \cdot {2 i} + {-7} \cdot {-3 i} + {22} \cdot {-3 i^2}} {13} $ Evaluate each product of two numbers. $ \dfrac{-14 + 44i + 21i - 66 i^2} {13} $ Finally, simplify the fraction. $ \dfrac{-14 + 44i + 21i + 66} {13} = \dfrac{52 + 65i} {13} = 4+5i $